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Published in:
2013 | OriginalPaper | Chapter
On ε-Uniform Error Estimates For Singularly Perturbed Problems in the DG Method
Author : V. Kučera
Published in: Numerical Mathematics and Advanced Applications 2011
Publisher: Springer Berlin Heidelberg
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Abstract
In this paper we present the analysis of the discontinuous Galerkin (DG) finite element method applied to a nonstationary nonlinear convection-diffusion problem. Using the technique of Zhang and Shu (SIAM J Numer Anal 42(2):641–666, 2004), originally for explicit schemes, we prove apriori error estimates uniform with respect to the diffusion coefficient and valid even in the purely convective case. We extend the cited analysis to the method of lines using continuous mathematical induction and a nonlinear Gronwall-type lemma. For an implicit scheme, we prove that there does not exist a Gronwall-type lemma capable of proving the desired estimates using standard arguments. Next, we use a suitable continuation of the implicit solution and use continuous mathematical induction to prove error estimates under a CFL-like condition.